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Mirrors > Home > MPE Home > Th. List > Mathboxes > recbothd | Structured version Visualization version GIF version |
Description: Take reciprocal on both sides. (Contributed by metakunt, 12-May-2024.) |
Ref | Expression |
---|---|
recbothd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
recbothd.2 | ⊢ (𝜑 → 𝐴 ≠ 0) |
recbothd.3 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
recbothd.4 | ⊢ (𝜑 → 𝐵 ≠ 0) |
recbothd.5 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
recbothd.6 | ⊢ (𝜑 → 𝐶 ≠ 0) |
recbothd.7 | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
recbothd.8 | ⊢ (𝜑 → 𝐷 ≠ 0) |
Ref | Expression |
---|---|
recbothd | ⊢ (𝜑 → ((𝐴 / 𝐵) = (𝐶 / 𝐷) ↔ (𝐵 / 𝐴) = (𝐷 / 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recbothd.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | recbothd.3 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | recbothd.4 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 0) | |
4 | 1, 2, 3 | divcld 11439 | . . . . . 6 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℂ) |
5 | recbothd.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0) | |
6 | 1, 2, 5, 3 | divne0d 11455 | . . . . . 6 ⊢ (𝜑 → (𝐴 / 𝐵) ≠ 0) |
7 | 4, 6 | jca 516 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 𝐵) ∈ ℂ ∧ (𝐴 / 𝐵) ≠ 0)) |
8 | recbothd.5 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
9 | recbothd.7 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
10 | recbothd.8 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ≠ 0) | |
11 | 8, 9, 10 | divcld 11439 | . . . . . 6 ⊢ (𝜑 → (𝐶 / 𝐷) ∈ ℂ) |
12 | recbothd.6 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 0) | |
13 | 8, 9, 12, 10 | divne0d 11455 | . . . . . 6 ⊢ (𝜑 → (𝐶 / 𝐷) ≠ 0) |
14 | 11, 13 | jca 516 | . . . . 5 ⊢ (𝜑 → ((𝐶 / 𝐷) ∈ ℂ ∧ (𝐶 / 𝐷) ≠ 0)) |
15 | 7, 14 | jca 516 | . . . 4 ⊢ (𝜑 → (((𝐴 / 𝐵) ∈ ℂ ∧ (𝐴 / 𝐵) ≠ 0) ∧ ((𝐶 / 𝐷) ∈ ℂ ∧ (𝐶 / 𝐷) ≠ 0))) |
16 | rec11 11361 | . . . 4 ⊢ ((((𝐴 / 𝐵) ∈ ℂ ∧ (𝐴 / 𝐵) ≠ 0) ∧ ((𝐶 / 𝐷) ∈ ℂ ∧ (𝐶 / 𝐷) ≠ 0)) → ((1 / (𝐴 / 𝐵)) = (1 / (𝐶 / 𝐷)) ↔ (𝐴 / 𝐵) = (𝐶 / 𝐷))) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → ((1 / (𝐴 / 𝐵)) = (1 / (𝐶 / 𝐷)) ↔ (𝐴 / 𝐵) = (𝐶 / 𝐷))) |
18 | 17 | bicomd 226 | . 2 ⊢ (𝜑 → ((𝐴 / 𝐵) = (𝐶 / 𝐷) ↔ (1 / (𝐴 / 𝐵)) = (1 / (𝐶 / 𝐷)))) |
19 | 1, 2, 5, 3 | recdivd 11456 | . . 3 ⊢ (𝜑 → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴)) |
20 | 8, 9, 12, 10 | recdivd 11456 | . . 3 ⊢ (𝜑 → (1 / (𝐶 / 𝐷)) = (𝐷 / 𝐶)) |
21 | 19, 20 | eqeq12d 2775 | . 2 ⊢ (𝜑 → ((1 / (𝐴 / 𝐵)) = (1 / (𝐶 / 𝐷)) ↔ (𝐵 / 𝐴) = (𝐷 / 𝐶))) |
22 | 18, 21 | bitrd 282 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) = (𝐶 / 𝐷) ↔ (𝐵 / 𝐴) = (𝐷 / 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ≠ wne 2949 (class class class)co 7143 ℂcc 10558 0cc0 10560 1c1 10561 / cdiv 11320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-po 5436 df-so 5437 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 |
This theorem is referenced by: lcmineqlem11 39591 |
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